I had a problem like yours with Slax. My problem was not a failure to detect the NIC, rather some dhcp failure. I did not see this error when booting Knoppix or Kanotix. I thought it might be an issue with how Slax was trying to obtain dhcp info from my router, but oddly enough I could boot Slax from my two other computers (same router) and it would dhcp just fine.
What I did on the problem computer was watch closely as the boot messages scrolled by. There was a call to "dhcpcd" with a few command line options that I can't remember off the top of my head. After it finished booting I did not have an IP address. I ran the exact same dhcpcd command I had observed during the boot process but added the -S option. This worked, and I had an IP address assigned. This manual step is required every time I boot Slax from this particular computer. Weird. I seem to remember that I could even run the identical boot dhcpcd command, without adding the -S option, and that still worked too. But I'm not 100% sure I'm remembering this correctly because I was working on multiple computers and might have confused myself as to which options worked on which computers. I haven't used Slax enough to have this little gotcha's details burned permanently into my brain. I just remember it was some problem with dhcpcd.